Two-scale convergence with respect to measures and homogenization of monotone operators

In 1989 Nguetseng introduced two-scale convergence, which now is a frequently used tool in homogenization of partial differential operators. In this paper we discuss the notion of two-scale convergence with respect to measures. We make an exposition of the basic facts of this theory and develope it in various ways. In particular, we consider both variableLpspaces and variable Sobolev spaces. Moreover, we apply the results to a homogenization problem connected to a class of monotone operators.

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A characterization of first order nonlinear partial differential operators on Sobolev spaces

Journal of Functional Analysis ◽

10.1016/0022-1236(80)90059-2 ◽

1980 ◽

Vol 38 (1) ◽

pp. 118-138 ◽

Cited By ~ 4

Author(s):

M Marcus ◽

V.J Mizel

Keyword(s):

Sobolev Spaces ◽

Differential Operators ◽

Partial Differential ◽

First Order ◽

Partial Differential Operators

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Inverse-monotone nonlinear differential operators of the second order

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500019740 ◽

1978 ◽

Vol 79 (3-4) ◽

pp. 193-226 ◽

Cited By ~ 1

Author(s):

Johann Schröder

Keyword(s):

Differential Operators ◽

Sufficient Conditions ◽

Monotone Operators ◽

Second Order ◽

Differential Inequalities ◽

Partial Differential ◽

Nonlinear Differential Operators ◽

Partial Differential Operators ◽

Ordinary Differential Operators

SynopsisThis paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

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On invertibility of linear partial differential operators in Hölder and Sobolev spaces

Sbornik Mathematics ◽

10.1070/sm2003v194n05abeh000736 ◽

2003 ◽

Vol 194 (5) ◽

pp. 733-744 ◽

Cited By ~ 1

Author(s):

O A Mitina ◽

V M Tyurin

Keyword(s):

Sobolev Spaces ◽

Differential Operators ◽

Partial Differential ◽

Partial Differential Operators ◽

Linear Partial Differential Operators

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The Calder´on-Zygmund Theory I: Ellipticity

10.23943/princeton/9780691162515.003.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Classical Theory ◽

Singular Integral ◽

Singular Integrals ◽

Differential Operators ◽

Integral Operators ◽

Singular Integral Operators ◽

Single Parameter ◽

Partial Differential ◽

Translation Invariant ◽

Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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